The Game of Tumbleweed is PSPACE-complete

TL;DR

This paper proves that determining the winner in the generalized Tumbleweed game is PSPACE-complete, using a novel reduction from a known PSPACE-complete Boolean formula game, highlighting the game's computational complexity.

Contribution

The paper introduces a generalized version of Tumbleweed and demonstrates its PSPACE-completeness through a novel embedding technique that simulates a Boolean formula game.

Findings

Deciding the winner in generalized Tumbleweed is PSPACE-complete.

The embedding technique simulates a non-planar game within a planar board.

The reduction uses a one-move race to enforce game protocol.

Abstract

Tumbleweed is a popular two-player perfect-information new territorial game played at the prestigious Mind Sport Olympiad. We define a generalized version of the game, where the board size is arbitrary and so is the possible number of neutral stones. Our result: the complexity of deciding for a given configuration which of the players has a winning strategy is PSPACE-complete. The proof is by a log-space reduction from a Boolean formula game of T.J. Schaefer, known to be PSPACE-complete. We embed the non-planar Schaefer game within the planar Tumbleweed board without using proper "bridges", that are impossible due to the board's topology. Instead, our new technique uses a one-move tight race that forces the players to move only according to the protocol of playing the embedded 4-CNF game.